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arXiv:1608.01794 · Journal of Physics A Mathematical and Theoretical

Incompatibility of quantum channels

Teiko Heinosaari, Takayuki Miyadera

Published Feb 9, 2017• 54 citations
arXivHTMLPDF

Abstract

Abstract Two quantum channels are called compatible if they can be obtained as marginals from a single broadcasting channel; otherwise they are incompatible. We derive a characterization of the compatibility relation in terms of concatenation and conjugation, and we show that all pairs of sufficiently noisy quantum channels are compatible. The complement relation of incompatibility can be seen as a unifying aspect for several important quantum features, such as impossibility of universal broadcasting and unavoidable measurement disturbance. We show that the concepts of entanglement breaking channel and antidegradable channel can be completely characterized in terms compatibility.

DOI
10.1088/1751-8121/aa5f6b

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Abstract Two quantum channels are called compatible if they can be obtained as marginals from a single broadcasting channel; otherwise they are incompatible. We derive a characterization of the compatibility relation in terms of concatenation and conjugation, and we show that all pairs of sufficiently noisy quantum channels are compatible. The complement relation of incompatibility can be seen as a unifying aspect for several important quantum features, such as impossibility of universal broadcasting and unavoidable measurement disturbance. We show that the concepts of entanglement breaking channel and antidegradable channel can be completely characterized in terms compatibility.
Abstract Both irreversibility and incompatibility are important features of quantum channels. Irreversibility of quantum channels characterizes the fundamental limitation in reconstructing information from their outputs. This concept has traditionally been studied for individual channels in a global sense. In this work we generalize the conventional definition of reversible channels to relatively reversible channels, where a quantum channel is reversible relative to another one; thus it is in a relative sense. Incompatibility refers to the impossibility of simultaneously implementing certain pairs of quantum channels, and lies at the very heart of quantum theory. By leveraging the concept of complementary channels, we obtain a direct connection between relative reversibility and channel compatibility. We further propose a quantifier of channel irreversibility in terms of incompatibility between the complementary channels and the identity channel. To illustrate and compare the quantifier of irreversibility with some other quantifiers in the literature, we evaluate them for some prototypical channels. Our results provide insights into the interplay between irreversibility and incompatibility, which may have potential applications in quantum error correction and the resource theory of incompatibility.
A pair of quantum channels is said to be incompatible if they cannot be realized as marginals of a single channel. This paper addresses the general structure of the incompatibility of completely positive channels with a fixed quantum input space and with general outcome operator algebras. We define a compatibility relation for such channels by identifying the composite outcome space as the maximal (projective) C*-tensor product of outcome algebras. We show theorems that characterize this compatibility relation in terms of the concatenation and conjugation of channels, generalizing the recent result for channels with quantum outcome spaces. These results are applied to the positive operator valued measures (POVMs) by identifying each of them with the corresponding quantum-classical (QC) channel. We also give a characterization of the maximality of a POVM with respect to the post-processing preorder in terms of the conjugate channel of the QC channel. We consider another definition of compatibility of normal channels by identifying the composite outcome space with the normal tensor product of the outcome von Neumann algebras. We prove that for a given normal channel, the class of normally compatible channels is upper bounded by a special class of channels called tensor conjugate channels. We show the inequivalence of the C*- and normal compatibility relations for QC channels, which originates from the possibility and impossibility of copying operations for commutative von Neumann algebras in C*- and normal compatibility relations, respectively.
A quantum channel is called incompatibility-breaking if it always destroys the incompatibility of quantum measurements. On the other extreme are incompatibility-preserving channels, which never destroy incompatibility. This work studies both types of channels through the mathematical framework of zonotopes and zonoids, an approach that provides a complete characterization of compatibility for dichotomic measurements and is particularly powerful in the qubit setting. Using the language of zonoids, we present necessary and sufficient conditions for when a qubit unital channel is incompatibility-breaking, which are also sufficient for nonunital channels. As one of our main results, we demonstrate the superactivation phenomenon of two incompatibility-breaking qubit channels that are no longer incompatibility-breaking when used in parallel. Translating this result into the task of quantum steering, it says that two copies of a two-qubit Werner state can be steered at noise threshold 1/21/21/2, even though this is impossible using just one copy. For the problem of incompatibility preservation, we prove that a quantum channel with matching input and output dimensions will preserve general incompatibility if and only if it is a unitary transformation. When the output dimension is larger, this result no longer holds, and we offer a conjectured form of a general incompatibility-preserving channel.
A typical bipartite quantum protocol, such as EPR-steering, relies on two quantum features, entanglement of states and incompatibility of measurements. Noise can delete both of these quantum features. In this work we study the behavior of incompatibility under noisy quantum channels. The starting point for our investigation is the observation that compatible measurements cannot become incompatible by the action of any channel. We focus our attention to channels which completely destroy the incompatibility of various relevant sets of measurements. We call such channels incompatibility breaking , in analogy to the concept of entanglement breaking channels. This notion is relevant especially for the understanding of noise-robustness of the local measurement resources for steering.
Quantum instruments describe outcome probability as well as state change induced by measurement of a quantum system. Incompatibility of two instruments, i. e. the impossibility to realize them simultaneously on a given quantum system, generalizes incompatibility of channels and incompatibility of positive operator-valued measures (POVMs). We derive implications of instrument compatibility for the induced POVMs and channels. We also study relation of instrument compatibility to the concept of non-disturbance. Finally, we prove equivalence between instrument compatibility and postprocessing of certain instruments, which we term complementary instruments. We illustrate our findings on examples of various classes of instruments.
A quantum observable and a channel are considered compatible if they form parts of the same measurement device, otherwise they are incompatible. Constrains on compatibility between observables and channels can be quantified via relations highlighting the necessary tradeoffs between noise and disturbance within quantum measurements. In this paper we shall discuss the general properties of these compatibility relations and then fully characterize the compatibility conditions for an unbiased qubit observable and a Pauli channel. The implications of the characterization are demonstrated on some concrete examples.